A PHASE TRANSITION IN THE COMING DOWN FROM INFINITY OF SIMPLE EXCHANGEABLE FRAGMENTATION-COAGULATION PROCESSES

• 成果类型: Article
• 署名作者: Foucart, Clement
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1691
• 发表日期: 2022
• 页码: 632-664
• 关键词: coalescents

摘要

We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a Lambda-coalescent, and fragmentation dislocates at a finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters theta(*) <= theta* is an element of [0, infinity], so that if theta* < 1, the process comes down from infinity and if theta(*) > 1, then it stays infinite. We illustrate our result with regularly varying coagulation and fragmentation measures. In this case, the parameters theta*, theta(*) coincide and are explicit.